Great idea, but these proposals tend to create turf battles between the airlines & the transit lines. In San Jose the local CalTrain service advertizes connections (via free shuttle bus) to the San Jose airport.
The San Francisco airport is getting a direct BART connection, but it took a long time to plan.
In Los Angeles the MTA Green Line passes a mile east and connects by free shuttle bus.
Newark NJ now has an AirTrain that connects to Amtrak's NEC/Penn Station Manhattan?
But people hate to schlepp a bunch of bags over several connecting modes so direct rail platform to air terminal transfers are best.
In Burbank the local Amtrak Surfliner stops about 1/4 mile-1/2 mile from the Air Terminal.
The trains help get some passengers to the airlines, but a lot of the airports facing cutbacks are in less populated areas and rail service would not be economical...